Related papers: On the regularity of primes in arithmetic progress…

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps…

Gross and Smith have put forward generalizations of Hardy - Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series that arises in these conjectures, up to lower order terms. More…

In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \geqq L $, there are at least $ N $ primes in the interval…

A notion of arithmetic similarity between number fields is defined by requiring equality of some arithmetic statistics over all but finitely many rational primes. The exceptional set is empty in all previously studied cases, but existing…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain…

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…

Given a variety over $\mathbb{Q}$, we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise…

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…

A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are…

Gerard and Washington proved that, for $k > -1$, the number of primes less than $x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes up to $x$. We extend this result to primes in arithmetic progressions: we prove…

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…

In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of primes $p \leq x$ for which the reductions of elliptic curves of small height modulo $p$ satisfy certain arithmetic properties, namely cyclicity and divisibility of…

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of…

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module…